The Jacobian of a regular orthogonal matroid and torsor structures on spanning quasi-trees of ribbon graphs

TL;DR

This paper extends the concept of Jacobian torsors from planar graphs to graphs on surfaces of arbitrary genus using orthogonal matroids, revealing new combinatorial structures and applications.

Contribution

It generalizes the torsor structure of spanning trees to spanning quasi-trees in ribbon graphs via orthogonal matroids, broadening the scope of previous planar graph results.

Findings

Generalization of Jacobian torsors to higher genus surfaces

Introduction of orthogonal matroids in graph combinatorics

Application of orthogonal matroids to graph theory problems

Abstract

Previous work of Chan--Church--Grochow and Baker--Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally forms a group, except that there is no distinguished identity element. We generalize this fact to graphs embedded on orientable surfaces of arbitrary genus, which can be identified with ribbon graphs. In this generalization, the set of spanning trees of $G$ is replaced by the set of spanning quasi-trees of the ribbon graph, and the Jacobian group of $G$ is replaced by the Jacobian group of the associated regular orthogonal matroid $M$ (along with an associated regular representation of $M$). Our proof shows, more generally, that the family of "BBY torsors" constructed by Backman--Baker--Yuen and later generalized by Ding admit natural generalizations to (regular representations of) regular orthogonal matroids. In addition to shedding light on the role of planarity in the earlier work mentioned above, our results represent one of the first substantial applications of orthogonal matroids (also called "even delta-matroids" or "Lagrangian orthogonal matroids") to a natural combinatorial problem about graphs.